Isotropy of Orthogonal Involutions

An orthogonal involution on a central simple algebra becoming isotropic over any splitting field of the algebra, becomes isotropic over a finite odd degree extension of the base field (provided that the characteristic of the base field is not 2). Our aim is a proof of the following result, generalizing the hyperbolicity statement of [5]: Theorem 1. Let F be a field of characteristic not 2, A a central simple F -algebra, σ an orthogonal involution on A. The following two conditions are equivalent: (1) σ becomes isotropic over any splitting field of A; (2) σ becomes isotropic over some finite odd degree extension of the base field. The proof of Theorem 1 is given in the very end of the paper; a sketch of the proof is given shortly below. For F with no finite field extensions of odd degree, Theorem 1 proves [8, Conjecture 5.2]. The general reference on central simple algebras and involutions is [11]. The implication (2) ⇒ (1) is a consequence of the Springer theorem on quadratic forms. We only prove the implication (1) ⇒ (2). Note that condition (2) is equivalent to the condition that σ becomes isotropic over some generic splitting field of the algebra, such as the function field of the Severi-Brauer variety of any central simple algebra Brauerequivalent to A. We prove this theorem over all fields simultaneously using an induction on the index indA of A. The case of indA = 1 is trivial. The case of indA = 2 is done in [15] (with “σ is isotropic (over F )” in place of condition (2)). From now on we are assuming that indA > 2. Therefore indA = 2 for some integer r ≥ 2. Let us list our basic notation: F is a field of characteristic different from 2; r is an integer ≥ 2; A is a central simple F -algebra of the index 2; σ is an orthogonal involution on A; D is a central division F -algebra (of degree 2) Brauer-equivalent to A; V is a right D-module with an isomorphism EndD(V ) ≃ A; v is the D-dimension of V (therefore rdimV = degA = 2 · v, where rdimV := dimF V/ degD is the reduced dimension of V ); we fix an orthogonal involution τ on D; h is a hermitian (with respect to τ) form on V such that the involution σ is adjoint to h; X = X(2; (V, h)) is the variety of totally Date: November 2009.